Waveguide bends and devices including waveguide bends

ABSTRACT

Waveguide bends are specially designed according to a matching condition in order to suppress mode distortion and other undesirable effects. The bend is structured having regard to its length and curvature to ensure that at its end the first and second bend modes are substantially in phase with each other having completed approximately an integer number of beats. By being in phase at the end of the bend, the two modes are able to properly reconstruct the first mode of the straight waveguide and propagate on with a minimum of distortion, whether it be into a straight section, a further curved section of arbitrary curvature, into a free space propagation region and the like. This approach suppresses mode distortion, transition losses and other negative effects of waveguide bends. Device applications include couplers, Y-branches and Mach-Zehnder interferometers, all of which include waveguide bends.

BACKGROUND OF THE INVENTION

[0001] The invention relates generally to waveguides including bends,more especially but not exclusively to optical waveguides, such asplanar waveguides, and to optical devices incorporating waveguides withbends.

[0002] Planar waveguides are used extensively in optical communicationsfor optical switching, multiplexing and demultiplexing. Many basicoptical components and devices, such as modulators, couplers andsplitters, incorporate waveguide bends. Moreover, to achieve highdensity of integration of optical devices on a single chip or substrate,it is often necessary to interconnect the devices with waveguide bends.The size and insertion loss of the waveguide bends ultimately determinethe maximum density of devices that can be integrated into an opticalcircuit of finite size.

[0003] It is well known that the field distribution of a fundamentalmode in a curved waveguide is different from that in a straightwaveguide. The field mismatch between the respective fundamental modesresults in transition losses and in excitation of higher-order modes atthe junctions between straight and curved waveguides.

[0004] In the prior art, one proposed solution to this problem is tooffset the curved waveguide at its junction with an adjoining straight(or curved) waveguide, with the offset being towards the centre ofcurvature of the curved waveguide [1-4]. A number of examples of suchprior art offset waveguide structures are now described.

[0005]FIG. 1 of the accompanying drawings shows a prior art structure asenvisaged by Kitoh et. al. [2] comprising a curved waveguide 10 with aconstant radius of curvature ‘r’ coupled to a straight waveguide 20. Thecentre line 14 of the curved waveguide 10 is laterally offset by adistance ‘d’ from the centre line 16 of the straight waveguide 20, withthe offset being towards the centre of curvature of the curved waveguide10.

[0006]FIG. 2 of the accompanying drawings shows an S-shaped waveguidedescribed by Kitoh et. al. [2] which may be considered as a combinationof two of the elements shown in FIG. 1. The S-shaped waveguide comprisesan input waveguide 24 and an output waveguide 26 that extends parallelto and laterally displaced from the input waveguide 24. The inputwaveguide 24 is coupled to a first curved waveguide section 28 which islaterally offset from the input waveguide by a distance ‘d’. The firstcurved waveguide section 28 is further coupled to a second curvedwaveguide section 30 of opposite curvature. The offset between the firstand second curved waveguide sections 28 and 30 is ‘2 d’. The secondcurved waveguide 30 section is further coupled to the output waveguide26. The offset between the second curved waveguide 30 and the outputwaveguide 26 is ‘d’. The first and second curved waveguide sections 28and 30 have the same radius of curvature.

[0007]FIG. 3 of the accompanying drawings shows a waveguide directionalcoupler described by Kitoh et. al. [2]. The waveguide directionalcoupler has four arms 47. Each arm 47 consists of a first straightsection 44 coupled to a first curved section 45 which is further coupledto a second curved section 46 which, in turn, is further coupled to astraight central section. The curved sections 45 and 46 are offset by adistance ‘2d’ to each other. The straight sections 44 and 49 are offsetto curved sections 45 and 46 respectively by distances ‘d’. Each curvedsection 45 and 46 has the same radius of curvature.

[0008] Reference [2] describes how the waveguide junction offsets can bedimensioned to minimize transition losses. However, with this transitionloss optimization, the offsets distort the optical field as it travelsbetween the straight and curved sections and excite a small amount ofradiative modes.

[0009] A different approach for optimizing junction offsets is taken inreferences [3] and [4] which describe how the waveguide junction offsetscan be dimensioned to minimize field distortion. However, with thisapproach the transition losses are higher.

[0010] Although the provision of offsets is effective in theory, inpractice it is demanding to fabricate waveguides with the proposedoffsets, since low dimensional tolerances are needed.

SUMMARY OF THE INVENTION

[0011] According to a first aspect of the invention there is provided awaveguide for guiding a field therealong, the waveguide comprising abend bounded by an input end and an output end, wherein the field in thebend is describable by first and second bend modes which are in phase atthe input end but propagate at different velocities through the bend,thereby coming out of phase and into phase with each other in beats,wherein the bend is structured having regard to its length and curvatureto ensure that at the output end the first and second bend modes aresubstantially in phase with each other over a desired wavelength rangeof the optical field having completed approximately an integer number ofbeats.

[0012] The basic structure of a waveguide bend is of course well known.However, the bend of the first aspect of the invention differs from aconventional waveguide bend in that there is a special shaping anddimensioning of the bend that allows the optical modes to propagatethrough the bend substantially undistorted.

[0013] The underlying idea behind the invention is to make a waveguidebend comprising one or more curved sections such that, at the end of thebend, the two orthogonal modes are in phase. By being in phase at theend of the bend, the two modes are able to properly reconstruct thefirst mode of the straight waveguide and propagate on with a minimum ofdistortion, whether it be into a straight section, a further curvedsection of arbitrary curvature, into a free space propagation region orwhatever. Typically, this is achieved by making a bend having a length‘1’ equal to an integer number of beat lengths L_(b). A waveguide bendor single curved portion that satisfies this condition is hereinafterreferred to as a matched bend or curved portion.

[0014] The utility of the invention is considerably enhanced by the factthat a matched bend is well matched over a relatively large wavelengthrange. This is because the beat length changes only very weakly withwavelength. This is quite surprising, since the propagation constants,coupling integral and other relevant mode parameters all vary withwavelength relatively strongly. However, although the mode parametersvary with wavelength, they do so in a very similar manner so that thenet result in terms of beat length variation is quite small. It is thuspossible to provide close matching of a bend for a large range ofwavelengths, which is of course a great advantage, for example forwavelength division multiplexed systems. For example, matching can beprovided across the whole of the third telecommunications window.

[0015] The proposed matched bend approach can be used to suppress modedistortion, transition losses and other negative effects of waveguidebends, without the need for inserting the small lateral offsets of theprior art [1-4]. Waveguide bends can thus be inserted into opticalcircuits where desired, e.g. to optimize packing density, withoutconcern for mode distortion or other negative effects on the componentsof the circuit adjacent to the bends.

[0016] In embodiments of the invention, the first and second bend modeshave a phase mismatch of less than one of 30, 20, 10, 5, 2 and 1 degreesat the output end of the curved portion over the desired wavelengthrange. The desired wavelength range may be any of 1, 2, 5, 10, 20, 30,40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180,190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 350, 400 and500 nm. These desired wavelength ranges may span any wavelength range atwavelengths of 300 nm to 2000 nm, more particularly any wavelengths of1000 to 1800 nm, still more particularly any wavelengths of 1200 to 1700nm.

[0017] In embodiments of the invention, the waveguide is a planarwaveguide, ridge waveguide, diffused waveguide, or any other waveguidefabricated using a substrate.

[0018] The invention may also find application in optical fiberwaveguides.

[0019] The invention is applicable to multimode as well as monomodewaveguides.

[0020] It is noted that the second bend mode may be leaky.

[0021] The integer number of beats may vary. Example values are 1, 2, 3,4 and 5.

[0022] The waveguide may be shaped to provide a U-bend or an L-bend. Ingeneral the bend may describe any desired angular deviation.

[0023] A design rule for a bend of arbitrary curvature characterized bya radius of curvature R that is a function of angle θ through the bendis such that the curved portion of the bend should substantially satisfythe following matching condition: $\begin{matrix}{{\int_{{- \theta}/2}^{\theta/2}{\sqrt{{{R^{2}(\theta)}C_{1}} + C_{2}}\quad {\theta}}} = {2N\quad \pi}} \\{where} \\{{C_{1} = \left( {\beta_{0} - \beta_{1}} \right)^{2}};} \\{{C_{2} = {4\beta_{0}\beta_{1}c_{10}^{2}}};}\end{matrix}$

[0024] β₀ & β₁ are respective phase constants of first and secondstraight modes;

[0025] c₁₀ is a coupling coefficient indicative of coupling induced bythe bend between the first and second straight modes; and

[0026] N is the integer number of beats.

[0027] It is noted that the invention is applicable to waveguide bendsnot only with varying bending radius, but also to waveguide bends whosewidth varies along the bend.

[0028] A design rule for one embodiment is valid in the case that thebend comprises an arcuate curved portion. Namely, where R is a constantover at least one arcuate part of the bend, the curved portion of thebend should substantially satisfy the following matching condition:$R = \sqrt{\frac{\left( \frac{2N\quad \pi}{\theta} \right)^{2} - C_{2}}{C_{1}}}$

[0029] The matching condition may be deemed to be substantiallysatisfied when N is within one of 10%, 8%, 6%, 4%, 2%, 1%, 0.5% and0.25% of an integer value over the desired wavelength range.

[0030] The matched bend may comprise multiple curves. The individualcurves may each themselves satisfy the matching condition.Alternatively, the individual curves may be unmatched, but collectivelyform a matched bend.

[0031] In the case of multiple curves these can be different in terms ofdimensions, shape or any other parameter relevant for the matching. Thisis easy to achieve based on the use of the design equations detailedherein. To give a concrete example, bend matching in an arrayedwaveguide grating should be possible using the theory presented herein,whereas this would be inconceivable to achieve otherwise. Descriptionsof arrayed waveguide gratings can be found in the literature, forexample in a number of US patents issued in the name of Dragone asinventor, the contents of which are incorporated herein by reference.

[0032] In one embodiment, the bend is an S-bend comprising two curvedportions of opposite curvature to provide a waveguide offset thatconnects two straight waveguide portions that extend parallel to eachother. A lateral offset is useful for placement between two opticalcomponents aligned substantially in parallel with each other.

[0033] A straight portion of waveguide can be defined as a waveguidehaving no bend with a radius of curvature of less than one of 20, 30,40, 50 and 100 cm.

[0034] The proposed matched bends may find application in a variety ofdevices. One example, is a Mach-Zehnder interferometer device comprisinga first arm and a second arm, wherein at least the first arm comprises amatched bend waveguide according to the first aspect of the invention.Another example is a waveguide branch, such as a Y-branch, comprising aninput waveguide and at least two output waveguides, wherein the inputwaveguide comprises a matched bend waveguide. A further example, is amultimode interference coupler having an input connected to a matchedbend waveguide. A still further example is an arrayed waveguide gratingin which at least some of the arms are matched bend waveguides. Yetanother example is an arrayed waveguide grating having an inputconnected to at least one matched bend waveguide. Another example is awaveguide coupler comprising first and second waveguides which extendproximal to each other to form a coupling region having first and secondends, wherein the first and second waveguides include curved portionswhich conform to the matched bend design rules, in order to diverge fromeach other at the first and second ends of the coupling region Moregenerally, further examples will be any optical circuit comprising anoptical component having an input for receiving an optical signal and amatched bend waveguide connected to the input. This will be especiallyuseful when the optical component is of a type that is sensitive to modedistortion at the input.

[0035] According to a second aspect of the invention, there is provideda method of manufacturing a waveguide for guiding an optical fieldtherealong, the waveguide comprising a bend bounded by an input end andan output end, the method comprising: describing the optical field inthe bend by first and second bend modes which are in phase at the inputend but propagate at different velocities through the bend, therebycoming out of phase and into phase with each other in beats; designingthe bend having regard to its length and curvature to ensure that at theoutput end the first and second bend modes are substantially in phasewith each other having completed approximately an integer number ofbeats; and fabricating the waveguide.

[0036] According to a third aspect of the invention, there is provided amode converter comprising a waveguide for guiding a field therealong,the waveguide having a bend bounded by an input end and an output end,wherein the bend is structured having regard to its length and curvatureto convert an optical field received at the input end with a predictableratio of a first power portion in a fundamental mode to a second powerportion in a first leaky mode into an optical field at the output end inwhich a substantially all the power is in the fundamental mode andsubstantially none in the first leaky mode. In practice it may besufficient that the percentage of power in the first leaky mode aftermode conversion is less than one of 10%, 5%, 2% and 1% of the power inthe fundamental mode.

BRIEF DESCRIPTION OF THE DRAWINGS

[0037] For a better understanding of the invention and to show how thesame may be carried into effect reference is now made by way of exampleto the accompanying drawings in which:

[0038]FIG. 1 is a schematic diagram of a first example waveguide fromthe prior art;

[0039]FIG. 2 is a schematic diagram of a second example waveguide fromthe prior art;

[0040]FIG. 3 is a schematic diagram of a third example waveguide fromthe prior art;

[0041]FIG. 4 is a schematic diagram of a first embodiment of theinvention showing a waveguide bend;

[0042]FIG. 5 is a cross-section through line A-A of FIG. 4;

[0043]FIG. 6 is a graph of radius of curvature R in meters against angleθ in degrees;

[0044]FIG. 7 shows a calculated field intensity pattern for a matchedbend waveguide according to the first embodiment;

[0045]FIG. 8 shows a calculated field intensity pattern for an unmatchedbend waveguide for comparison with FIG. 7;

[0046]FIG. 9 is a schematic diagram of a second embodiment of theinvention showing an S-bend waveguide offset;

[0047]FIG. 10 shows a calculated field intensity pattern for a matchedwaveguide offset according to the second embodiment;

[0048]FIG. 11 shows a calculated field intensity pattern for anunmatched waveguide offset for comparison with FIG. 10;

[0049]FIG. 12 is a schematic diagram of a waveguide couplerincorporating S-bend offsets of the kind shown in FIG. 9;

[0050]FIG. 13 is a schematic diagram of a third embodiment of theinvention showing a Y-branch preceded by a waveguide offset;

[0051]FIG. 14 shows a calculated field intensity pattern for a matchedbend structure according to the third embodiment;

[0052]FIG. 15 shows a calculated field intensity pattern for anunmatched bend structure for comparison with FIG. 14;

[0053]FIG. 16 is a block schematic diagram of a multimode interference(MMI) coupler according to a fifth embodiment of the invention;

[0054]FIG. 17 shows a calculated field intensity pattern for a matchedbend MMI coupler according to the fifth embodiment;

[0055]FIG. 18 shows a calculated field intensity pattern for anunmatched bend MMI coupler for comparison with FIG. 17;

[0056]FIG. 19 is a schematic diagram of a 1×4 power splitter based oncascaded multimode interference (MMI) couplers according to a sixthembodiment of the invention;

[0057]FIG. 20 is a schematic diagram of a Mach-Zehnder filter deviceaccording to a seventh embodiment of the invention;

[0058]FIG. 21 is a schematic diagram of an arrayed wave guide grating(AWG) device according to an eighth embodiment of the invention;

[0059]FIG. 22 is a schematic diagram of a mode converter according to aninth embodiment of the invention; and

[0060]FIG. 23 is a schematic diagram of an example mode converteraccording to the ninth embodiment.

DETAILED DESCRIPTION First Embodiment

[0061]FIG. 4 shows a plan view of a monomode waveguide 50 according to afirst embodiment of the invention. The waveguide comprises a core andcladding. The waveguide is made up of a first straight section 60extending in direction ‘x’ followed by a curved section 58 followed by asecond straight section 55. The curved section 58 has a radius ofcurvature R. The radius of curvature R is measured to a centre line ofthe core, which is shown with a dashed line in the figure. The curvedsection 58 extends through an angle θ having a circumferential length 1,as measured along the centre line. The illustrated waveguide thus has anarcuate bend. In one example, R=2 mm, θ=5.5° and 1=191 μm.

[0062] The basic structure of a waveguide bend is of course well known.However, the bend according to the first embodiment of the inventiondiffers from a conventional waveguide structure in that there is aspecial shaping and dimensioning of the bend that allows the opticalmodes to propagate from the first straight section into the secondstraight section 55 substantially undistorted. The design rules forshaping and dimensioning the bend are described in detail further below.

[0063]FIG. 5 is a cross-section through A-A of FIG. 4. As is evident,the monomode waveguide is realized as a ridge waveguide with the core 52arranged as a ridge on top of the cladding 54. A cap layer 53 is alsoprovided on top of the core 52. (The cap layer may also be referred toas a cover layer or upper clad layer). An one example, the core 50 has awidth w=8.5 μm and a height h=6 μm, and the cap layer has a thicknessc=4 μm. If desired, the additional cap layer may be omitted. It willalso be understood that illustration of a ridge waveguide is by way ofexample only. The waveguide may be buried, diffused or of any otherkind.

[0064] The structure may be fabricated by any of several knowntechniques, for example sol-gel, ion-beam implantation, vacuumdeposition, implantation, chemical or physical vapor deposition or iondiffusion. The core 52 is made from doped silica with refractive indexn=1.4652. The cladding 54 and cap 53 is made from silica and has arefractive index n=1.4552. The refractive index difference between coreand cladding Δn is thus Δn=0.69%.

[0065] The materials chosen for the core and cladding may be of anydielectric material with refractive indices that will allow light to beguided through the waveguide.

[0066] The design rules for shaping and dimensioning the bend of thefirst embodiment (and also waveguide bends of other embodiments) are nowdescribed. It has been shown [5, 6] that the modes of a bent waveguidecan be approximated as a linear combination of the straight waveguidesmodes. For a monomode waveguide, it can be demonstrated that only thefundamental mode and the first leaky mode in the plane of the bend (bothof the straight waveguide) are sufficient to describe the fundamentalbend mode. Vice versa, the fundamental straight mode can be described asthe linear combination of the fundamental bent mode and the first leakybend mode. This is valid, separately both for TE and TM polarization.

[0067] When a straight fundamental mode enters in a bend you can imaginethis mode as the sum of two modes, the two bend modes (fundamental+firstleaky). These two modes are not coupled (because they are the modes ofthe bend waveguide) but propagate at different velocities in the bend.The two modes have different phase constants, β₀ and β₁, related to β₀and β₁, the phase constants of the two straight modes, by some complexrelations (see [5, 6]). This causes the total optical field to changeshape as it travels through the curved waveguide section. At thebeginning of the curve, the two modes are in phase with each other, butgradually dephase. At some point along the curve, provided it issufficiently long, the two modes come back into phase with each otheragain. In other words, the phase difference reaches 2π Thecharacteristic length of curved waveguide section over which a 2π phasedifference will occur is determined by the difference in the phasevelocities of the two modes. This length is referred to here as the beatlength L_(b). As the two bend modes propagate along the curve, they willthus come into phase periodically, after each beat length ofpropagation. Therefore, at each integral number of beat lengths, theoptical field in the curved waveguide section will momentarilyreconstruct to have the form of the optical field of the fundamentalmodes in a straight waveguide section, provided that the radiationlosses of the leaky bend mode are not too high, i.e. provided that thebending radius is not too small.

[0068] The underlying idea behind the invention, not just the presentembodiment, is to make a curve such that, at the end of the curve, thetwo orthogonal modes are in phase. By being in phase at the end of thecurve, the mode is able to properly reconstruct the first mode of thestraight waveguide and propagate on with a minimum of distortion,whether it be into a straight section, a further curved section ofarbitrary curvature, a free space propagation region or whatever.Typically, this is achieved by making a curve having a length ‘1’ equalto an integer number of beat lengths L_(b). A curved waveguide sectionthat satisfies this condition is hereinafter referred to as a matchedbend.

[0069] This condition is now quantified in the case of an arcuate curvehaving a single radius of curvature R, constant width W and describingan arc of angle θ, such as that illustrated in FIG. 4. In this case, thebeat length is: $\begin{matrix}{L_{b} = \frac{2\pi}{\beta_{B0} - \beta_{B1}}} & (1) \\{where} & \quad \\{\left( {\beta_{B0} - \beta_{B1}} \right) = {\frac{1}{R}{\sqrt{{R^{2}\left( {\beta_{0} - \beta_{1}} \right)}^{2} + {4\beta_{0}\beta_{1}c_{10}^{2}}}.}}} & (2)\end{matrix}$

[0070] as will be understood from references [5, 6] and where

[0071] β₀ & β₁ are the phase constants of the first two modes of thestraight waveguide; and

[0072] c₁₀ is the coupling coefficient between the first two modes ofthe straight waveguide induced by the curve.

[0073] To obtain a curve containing exactly N beat lengths, that is thematched bend condition, the condition is imposed that:

Rθ=NL_(b)  (3)

[0074] and taking into account equations (1) and (2) the optimum bendingradius is found to be $\begin{matrix}\begin{matrix}{R = \sqrt{\frac{\left( \frac{2N\quad \pi}{\theta} \right)^{2} - C_{2}}{C_{1}}}} \\{where} \\{{C_{1} = \left( {\beta_{0} - \beta_{1}} \right)^{2}};} \\{{C_{2} = {4\beta_{0}\beta_{1}c_{10}^{2}}};{and}}\end{matrix} & (4)\end{matrix}$

[0075] N is a positive integer that indicates the length of the curve inunits of beat length.

[0076] In the more general case of a curve with variable bending radiusR=R(θ), and/or a waveguide bend with varying width W=W(θ), the sameprocedure leads to an integral equation that relates R to θ:$\begin{matrix}{{\int_{{- \theta}/2}^{\theta/2}{\sqrt{{{R^{2}(\theta)}{C_{1}(\theta)}} + C_{2}}\quad {\theta}}} = {2N\quad {\pi.}}} & (5)\end{matrix}$

[0077] where the dependency of C₁ and C₂ on=θ follows from takingaccount of the possibility of variable waveguide width, since thecoefficients β₀, β₁ and c₁₀ are waveguide width dependent.

[0078]FIG. 6 is a graph showing results for a waveguide as shown in FIG.5, with parameter values: c₁₀=1.4533·10⁻⁶; β₀=5.9185·10⁶ andβ₁=5.8890·10⁶ (at a wavelength of 1550 nm). The graph refers to a bendwith a constant bending radius R and constant waveguide width W. Thegraph plots sets of values of the radius of curvature R in meters andangle of curvature θ in degrees that satisfy equation (1) above. Thegraph includes four traces, one for each of N=1, 2, 3 and 4. The tracefor N=1 indicates a set of values of R & θ that give a length of curvedwaveguide section equal to one beat length L_(b). The traces for N=2, 3and 4 indicate further data sets of R & θ for which the curved waveguidesections have lengths 1=2L_(b), 3L_(b) and 4L_(b) respectively.

[0079] In one example N=1, R=2 mm, θ=5.5°, and 1=191 μm. The position ofthis example on the graph of FIG. 6 is marked with a cross. It is notedthat the same angle can be obtained by a bending radius of 4 mm and alength 1=382 μm (two L_(b), N=2) or more (N=3,4. . . ).

[0080]FIGS. 7 and 8 show calculated mode patterns for two single bendshaving the structure shown in FIG. 4. More especially, the graphs showthe intensity of the calculated optical field propagating in the bend.In the case of FIG. 7, the bend conforms to the matched bend conditionset out above. For comparison, the bend of FIG. 8 is for an unmatchedbend. The matched bend of FIG. 7 has R=2 mm, θ=5.5° and 1=191 μm (N=1).The unmatched bend of FIG. 8 has R=2 mm, θ=2.75°, 1=95 Mm (N=0.5).

[0081] For the matched bend of FIG. 7, some slight mode distortion isevident as the mode propagates through the curved section. However,after the end of the curved section, the mode is seen to reconstructimmediately, substantially without distortion, in the second straightsection. In other words, the field pattern in the second straightsection reproduces the field pattern in the first straight section, asdesired.

[0082] For the unmatched bend of FIG. 8, by contrast, the mode does notreconstruct after the end of the curved section, at least not forseveral hundred microns of propagation included in the plot. The modecan be expected to reconstruct eventually, provided that the secondstraight waveguide mode radiates away completely. In addition, theunmatched bend results in significant radiation losses. For the presentexamples, the radiation losses of the matched bend are equal to 0.5%(0.02 dB) while those of the unmatched bend are 9.2% (0.42 dB).

[0083] In summary, the high mode distortion in and subsequent to theunmatched bend is associated with energy losses, which are of coursehighly undesirable for many applications. By contrast, the lowdistortion of a matched bend is associated with low transition losses inthe bend.

[0084] Perhaps more importantly, the slow recovery of the mode after anunmatched bend can have a severe impact on the behavior or functionalityof an optical component placed after an unmatched bend. Some types ofoptical component are highly sensitive to input mode distortion, forexample splitters and components using splitters. To increaseintegration density it is desirable to pack components close together.However, slow recovery of the mode after a bend will mitigate againstachieving high packing density, since a large distance needs to bemaintained between the end of the bend and the next optical component,e.g. 500-1000 μm or even more. By contrast, the rapid mode stabilizationof a matched bend allows subsequent mode-sensitive optical components tobe placed close to the bend, e.g. within 100 μm or 200 μm, or even less,without problems.

[0085] Another strength of the proposed design is that it is relativelyinsensitive to wavelength variations. In other words a matched bend iswell matched over a relatively large wavelength range. This is explainedas follows. The bending radius R and the angle θ of an optimum benddepend on the beat length L_(B) defined in equation 1 through equations4-7, i.e. on the difference between the phase constants β_(B0 b) andβ_(B1) of the bend modes. Although β_(B0) and β_(B1) as well as β₀ and01 and the coupling integral c₁₀ depend on wavelength, they do so in avery similar manner so that the wavelength dependence of the beat lengthL_(B) is extremely weak. As a result, an optimum bend, offset or otherwaveguide structure is well matched over a broad wavelength region of100 nm or more in the second or third telecommunications windows (1.3and 1.55 μm bands). For example, it is possible to provide good matchingover the whole of the third telecommunications window.

Second Embodiment

[0086]FIG. 9 shows a plan view of a monomode waveguide 50 according to asecond embodiment of the invention. The waveguide 50 comprises a core 52and cladding 54. The waveguide 50 is made up of a first straight section60 extending in direction ‘x’ followed by a first curved section 58followed by a second curved section 56 of opposite curvature directionfurther followed by a second straight section 55 also extending in thedirection ‘x’. The first curved section 58 has a radius of curvature R₁.The second curved section 56 has a radius of curvature R₂=R₁=R. In eachcase, the radius of curvature is measured to a centre line of the core52. The curved sections 56 and 58 extend through the same angle θ suchthe the respective lengths of the curved sections 56 and 58 are equal.Together the curved sections 56 and 58 make up an S-bend 70. The S-bend70 provides a lateral offset ‘f’ in a direction ‘y’ perpendicular to ‘x’between the two straight sections of waveguide 60 and 55 In a specificexample, f=20 μm, as measured between the respective centre lines of thewaveguide straight sections 55 and 60.

[0087] The basic structure of an S-bend is well known. However, theS-bend 70 according to the second embodiment of the invention differsfrom a conventional waveguide structure in that there is a specialshaping and dimensioning of the bend that allows the optical modes topropagate from the first straight-section 60 into the second straightsection 55 substantially undistorted. The design rules for shaping anddimensioning the bend are described in detail further below.

[0088] The waveguide may be a ridge waveguide as shown in FIG. 5 inrelation to the first embodiment. Specifically, FIG. 5 is across-section through A-A of FIG. 9 in one example. The same parametervalues and variations are envisaged, as discussed in relation to thefirst embodiment.

[0089] The design rules for shaping and dimensioning the S-bend of thesecond embodiment both for unsymmetrical offset, and also the offsetwith variable bending radius, are derivable from equations (4) and (5).

[0090] For a given offset f, with the equations (1) to (3) and sometrigonometry, the couple R, θ is given by $\begin{matrix}\left\{ \begin{matrix}{R = {\frac{1}{2C_{1}}\left( {\frac{\left( {2N\quad \pi} \right)^{2}}{f} + \sqrt{\frac{\left( {2N\quad \pi} \right)^{4}}{f^{2}} - {4C_{1}C_{2}}}} \right)}} \\{\theta = \sqrt{\frac{f}{R}}}\end{matrix} \right. & (6)\end{matrix}$

[0091] and the offset is long then L=2{square root}{square root over(fR)}.

[0092] The generalization to an offset containing bends of variablebending radius and/or waveguide width is straightforward, and similar toequation (5).

[0093] If, instead, the length of the offset if fixed to L, theparameter pair R, θ is given by $\begin{matrix}\left\{ \begin{matrix}{R = \sqrt{\frac{C_{2}}{\left( \frac{4N\quad \pi}{L} \right)^{2} - C_{1}}}} \\{\theta = \frac{L}{2R}}\end{matrix} \right. & (7)\end{matrix}$

[0094] and the obtained offset is f=L²/4R.

[0095] In a specific example of an offset between two straightwaveguides: R=5 mm, θ=2.26°, N=1, 1=197 μm, L=395 μm, f=7.8 μm. Theexample also assumes the waveguide dimensions and other parameters givenfor the specific example of FIG. 5. The working point is indicated witha circle in FIG. 6.

[0096] The offset in a specific example with unmatched bends isidentical, but N=2, θ=3.0°, L=523 μm.

[0097]FIGS. 10 and 11 are graphs showing the intensity of the calculatedoptical field propagating in the two example offsets, FIG. 10 being thematched S-bend and FIG. 11 the unmatched S-bend.

[0098] The results are similar to those for the first embodiment inthat, for the offset in which the two bends of the S-bend are matched,the mode is seen to reconstruct immediately, substantially withoutdistortion, in the second straight section. On the other hand, for theunmatched structure, the mode is severely distorted subsequent to thecurved sections. Similar comments apply as made in relation to the firstembodiment. In respect of radiation losses, the offset with matchedbends has losses equal to 0.2% (0.009 dB) while the losses of the offsetwith unmatched bends are 6.2% (0.28 dB).

[0099] An important generalization of this embodiment is now discussed.All that is important for matching the S-bend structure, or any bendstructure comprising multiple curves, is that at the end of the bendstructure the first and second bend modes are substantially in phasewith each other having completed an integer number of beats. In thecontext of the twin-curve structure of the second embodiment, this meansthat it is not important whether the first and second bend modes are inphase at the end of the first curved section 58, but only important thatthey are in phase with each other at the end of the bend, i.e. at theend of the second curved section 56. In other words, a matched S-bendcan be made up of two appropriately designed “unmatched” curves, insteadof two matched curves, as described above. In this respect the two“unmatched” curves may be of different shape (eg. different bendingradius) or width (e.g. each with variable widths that are different).Moreover, the input and output straight sections need not be parallel toeach other.

Third Embodiment

[0100] An example of a device that incorporates waveguide bends oroffsets is a waveguide coupler. A waveguide coupler may be fabricated byplanar, ridge or diffused waveguides for example.

[0101]FIG. 12 shows a four-port (2×2) waveguide coupler according to athird embodiment of the invention. The coupler comprises first andsecond waveguides 80 and 82 which approach each other and run alongsideeach other over a coupling region 84 of length L_(C) over which modes inthe adjacent waveguides can exchange power through evanescent fieldinteraction. The length of the coupling region 84 determines whatproportion of input light power P₀ travelling in the first waveguide 80at the start of the coupling region 84 remains in the first waveguide P₁or is transferred into the second waveguide P₂ at the end of thecoupling region.

[0102] A waveguide coupler fabricated on a substrate (as opposed to afiber coupler) will require waveguide bends at each end of the couplingregion. In the illustrated structure, the coupling region is bounded bywaveguide offsets 70 according to the second embodiment, there beingfour in total, two for each waveguide. The design of the offsets canfollow the design rules set out for the second embodiment, even if thewaveguide portions forming the offset are partially co bed to eachother, as will be the case for the waveguide sections immediatelyadjacent to the coupling regions. (Alternatively, single bends or othermultiple bend waveguide shapes could be used).

[0103] At one level, the basic power division operation of the coupleris not strongly affected by the waveguide bends bounding the couplingregion, since the coupling coefficient (ratio P₂/P₀) is relativelyinsensitive to whether the waveguide bends of the offset are matched orunmatched. However, the higher losses and mode distortion discussed withreference to the previous embodiments will occur. Regarding modedistortion, as discussed further above, and also in reference [6], anunmatched bend results in excitation of a leaky mode (first order mode)which propagates through the coupling region and into the output,causing distortion of the output mode. This will potentially causeproblems with subsequent devices, such as a Y-branch, positioned afterthe coupler.

Fourth Embodiment

[0104]FIG. 13 shows a device according to a fourth embodiment. Thedevice is a Y-branch 86 preceded by a matched S-bend offset 70, that isan S-bend that satisfies the above-described matching condition. AY-branch is used for splitting the power P₀ carried in an inputwaveguide into two output waveguides which carry respective powers P₁and P₂. Typically, the Y-branch will be designed to split the powerequally, that is so that P₁=P₂.

[0105] In the illustrated device, the bends of opposite curvature extendover a distance L and are followed by a straight waveguide portion 87 oflength L₁ which in turn terminates in a waveguide split 88 from whichtwo waveguides 81 and 83 emerge, each extending from the split 88 withan S-bend offset of length L. These offsets are dimensioned to result inthe two waveguides emerging from the device separated by a lateralseparation d. For the sake of simplicity, the offset lengths, bendingradii and bend angles are shown to be the same (L, R and θ) for thevarious offsets and bends, but of course this will not generally be thecase.

[0106]FIG. 14 shows a calculated optical field pattern for an example ofsuch a Y-branch. The dimensions of the Y-branch in the example are:L=1.5 mm, waveguide separation d=30μm, R=5 cm, θ=0.5°. In the example,the waveguides are Ti diffused lithium niobate monomode waveguides withΔn=0.2%, 6 μm width and 3.31 μm height. The offsets are made up of twobends with R=5 cm, θ=0.5° (matched, N=1). The distance between the endof the offset and the Y-splitting point is L₁=700 μm.

[0107] In the figure, the vertical axis shows propagation direction xand the horizontal axis an orthogonal direction y, both axes being inunits of microns. As is evident from the figure, the optical fields inthe matched S-bend and the subsequent straight waveguide section at theinput are undistorted. This results in undistorted and equal opticalfields in the two output waveguide sections. Moreover, although notimmediately apparent from the figure, the results show that the outputpower-coupled into each of the output waveguide sections is almostidentical. Moreover, full stabilization of the mode power occurs shortlyafter the Y-split.

[0108]FIG. 15 is included for comparison with FIGS. 14. The resultsshown are for a Y-branch using similar parameters to the example of thefourth embodiment, but with the offsets being made up of unmatched bendswith θ=0.25° (N=0.5).

[0109] As is evident from FIG. 15, the optical fields in the unmatchedS-bend and the subsequent straight waveguide section at the input aredistorted. This results in distorted and unequal optical fields in thetwo output waveguide sections. Moreover, although not immediatelyevident from the figure, the results also show that the output powercoupled into each of the output waveguide sections differssubstantially.

[0110] The performance of the example Y-branches with matched andunmatched end offsets is summarized in Table I below.

[0111] Referring to the first row of results, when the Y-branch is madewith matched bends and preceded by a matched bend offset, the splittingratio is slightly asymmetrical and the losses are not zero, butrelatively low.

[0112] Referring to the second row of results, when an unmatched bendY-branch is preceded by an unmatched bend offset, a large powersplitting imbalance occurs. Losses also increase slightly. TABLE I P₁ P₂[%] [%] Imbalance Loss Y-branch with 0.500 0.494 1.17% (0.05 dB) 0.6%(0.026 dB) matched offset Y-branch with 0.565 0.428 32.2% (1.2 dB) 0.7%(0.03 dB) unmatched offset

[0113] In summary, if a Y-branch is to be preceded by a waveguideoffset, or other waveguide bend structure, matching the bends accordingto the design rules set out above can be of vital importance.

Fifth Embodiment

[0114]FIG. 16 shows a fifth embodiment of the invention comprising a 1×2multimode interference (MMI) coupler 92 preceded by an S-bend 72 similarto that of the second embodiment. The purpose of this MMI couplerembodiment is to provide a specific example of a device that issensitive to input mode distortion. A straight input waveguide section94 interconnects the S-bend 72 with the 1×2 MMI coupler 92. At theoutput side of the MMI coupler 92 there are provide first and secondstraight output waveguide sections 96 and 100. The-coupling into thestraight output waveguide sections 96 and 100 is realized by symmetricinterference, as described in reference [7]. The MMI coupler has alength ‘c’ and a width ‘b’.

[0115] In an example, the waveguide sections 72, 94, 96 and 100 areburied waveguide sections having a core width w=5.2 μm and a heighth=5.2 μm. The refractive index difference between the core and claddingΔn=0.69%. The MMI coupler 92 has a length c=1595 μm and a width b=30 μm.The first straight output waveguide section 96 is placed 8 μm above thecentre line of the MMI coupler 92 (as defined by the input waveguidesection 94). The second straight output waveguide section 100 is placed8 μm below the centre line of the MMI coupler 92. The S-bend 72comprises two matched curves conforming to equation (1) above. Thestraight section 94 has a length 0 82 m, i.e. the MMI coupler 9 followsimmediately from the end of the waveguide bend 72. Each curve has aradius of 10 mm and extends through an angle of 1.62°.

[0116] With the device, the output powers P1 and P2 coupled into the twooutput waveguide sections 96 and 100 were calculated, as well as thetotal transition loss through the coupler and preceding input waveguidesections. For perfect performance, the respective output powers shouldbe equal. In other words a 50:50 power split should be achieved at theoutput.

[0117] By way of comparison with the matched bend example, results werealso calculated for a device similar to that of FIG. 16, but with asimple straight input section instead of an S-bend input section(control), and a device similar to that of FIG. 16 but with an unmatchedS-bend in the input section (unmatched S-bend). The unmatched S-bend hasa bend radius R=10 mm and extends through an angle θ=0.8°. The resultsare given in Table II below. TABLE II P1 P2 Imbalance Loss [%] [%][%/dB] [%/dB] Control 0.4537 0.4536  0.02%  9.3% (0.42 dB) Matched0.4543 0.4534  0.2%  9.2% S-bend (0.42 dB) Unmatched 0.5305 0.3485 52.2%12.1% S-bend (1.8 dB) (0.56 dB)

[0118] For the control example, when light enters the MMI couplerthrough a straight waveguide alone, the input power is equally dividedbetween the two straight output waveguides, as expected.

[0119] For the example with a matched S-bend on the input side of theMMI coupler, the splitting ratio between the two output waveguidessections is almost exactly 50:50. In other words, the presence of thematched S-bend once input has not significantly affected the splittingratio of the MMI coupler. The total loss is somewhat higher as a resultof the S-bend.

[0120] For the example with an unmatched S-bend on the input side of theMMI coupler, the power splitting ratio is greatly affected by the modedistortion at the input. There is an imbalance of 1.8 dB (i.e. 52%)between the power coupled into the two output waveguides sections. Thetotal loss is also somewhat higher than for the matched S-bend example.

[0121]FIG. 17 is a calculated optical field intensity pattern for theexample of the fifth embodiment. As is evident from the figure, the modeis relatively undistorted at the input to the MMI coupler (0<x<500 μm).The mode pattern is rather complex within the MMI coupler (500<x<2100μm). At the output waveguide sections (x>2100) the modes are relativelyundistorted already after only about 100 μm of propagation in the outputchannels. The power balance between the two channels is also even(although not immediately apparent from the figure). Namely, after apropagation distance of 2150 μm, i.e. a short distance into the straightoutput waveguides, the power split is nearly equal.

[0122]FIG. 18, by way of comparison, is a calculated optical fieldintensity pattern for an MMI coupler similar to that of the example, butpreceded by an unmatched S-bend. As is evident from the figure, the modeis somewhat distorted at the input to the MMI coupler (0<x<500 μm). Themode pattern is rather complex within the MMI coupler (500<x<2100 μm).At the output waveguide sections (x>2100) the modes are highly distortedand a large power imbalance is also evident. The relative powerimbalance is approximately 0.53/0.35.

Sixth Embodiment

[0123]FIG. 19 is a block schematic drawing of a sixth embodiment of theinvention. The sixth embodiment is a 1×4 power splitter realized bycascading three 1×2 MMI couplers 202, 204 and 206 in two stages. A firstMMI coupler 202 has a straight input waveguide 94. The first MMI coupler202 has two output waveguides in the form of matched S-bends 112. TheS-bends 112 extend to form respective inputs for second and third MMIcouplers 204 and 206. The second MMI coupler 204 has two straight outputwaveguide sections that carry output powers PI and P2 in use. The thirdMMI coupler has two straight output waveguide sections that carry outputpowers P3 and P4 in use.

[0124] The waveguide and MMI coupler dimensions are the same as for thefifth embodiment described above. The bend radius R of the two S-bends112 that connect the MMI couplers 202, 204 and 206 is R=20 mn and theangle of curvature of the curved sections is θ=0.8°.

[0125] The calculated power imbalance between the four output powersPI-P4 is only 0.006 dB or 0.1%.

[0126] By comparison, in an example in which the S-bends interconnectingthe first and second stages of the device are unmatched (R=20 mm,θ=0.4°), the calculated power imbalance is much higher, namely 0.61 dBor 15.1%.

[0127] It is thus evident that power splitters based on MMI couplers aresensitive to optical field distortions so that the provision of matchedbends greatly improves performance. The improvement becomes moresignificant as the number of stages of such a power splitter increases.For example, the benefit of providing matched bends becomes greater asone progresses from a three stage 1×8 splitter, to a four stage 1×16splitter, a five stage 1×32 splitter or, more generally, to an n-stage1×2^(n) splitter.

[0128] Similar results are found for single stage or multi-stagecascaded Y-branch power splitters, i.e. an n-stage 1×2^(n) splitter.

[0129] In cascaded Y-branch or MMI coupler devices implemented in planarwaveguide technology, bends cannot usually be avoided, so the provisionof matched bends interconnecting the different stages of the cascadeprovides a major improvement, especially for multiple stage devices.

Seventh Embodiment

[0130]FIG. 20 relates to a seventh embodiment of the invention showing aMach-Zehnder filter comprising first and second MMI couplers 93 and 94having straight input and output waveguides 91 and 95 interconnected byfirst and second waveguide sections 132 and 134 of different opticalpath length which constitute first and second arms of the Mach-Zehnderfilter. The first arm 132 is bent to increase its optical path lengthand thus introduce the optical path difference between the two arms.More specifically, the first arm 132 comprises four curved waveguidesections. The waveguides and MMI couplers are as for the fifthembodiment.

[0131] In an example of this embodiment, the curved sections of thefirst arm each have a constant radius or curvature R=7 mm and extendthrough an angle θ=2.3° which complies with the matching condition ofequation (1). The waveguides are buried waveguides with corecross-section dimensions of 5.2 μm square and core/clad refractive indexdifference Δn=0.69%. In this case the extinction ratio of the device is42 dB. Extinction ratio is a measure of goodness of a Mach-Zehnderfilter.

[0132] By way of comparison, if the curves are unmatched with R=7 mm andθ=1.3°, the extinction ratio worsens to 21 dB.

[0133] If the angle of curvature is increased beyond the optimum matchedvalue of θ=2.3° for R=7 mm, the extinction ratio deteriorates evenfurther.

[0134] Similar results are obtained when Y-branches are used in place ofMMI couplers in a Mach-Zehnder filter, and for Mach-Zehnder modulators.

[0135] In simulations of an example of this embodiment, there is somevariation in the output power with propagation distance, but thevariation is less than 10%. By comparison, simulations of a similardevice with unmatched bends show an output intensity that wildlyoscillates with propagation distance. This difference in performance isattributed to the different levels of distortion in the mode enteringthe second MMI coupler 94 from the first (bent) arm 132.

[0136] Finally, it is noted that similar results are found for aMach-Zehnder interferometer fabricated with input and output sideY-branches, and for a device in which the MMI couplers are replaced withwaveguide couplers.

Eighth Embodiment

[0137]FIG. 22 is a schematic plan view of an active arrayed waveguidegrating (AWG) according to an eighth embodiment of the invention. Thedevice comprises an array of waveguides 142 interconnecting first andsecond free space propagation regions 148 and 146, successive waveguidesof the array having an incrementally increasing optical path length, asis known in the art, for example reference [4]. In contrast to aconventional AWG, the waveguides of the array 142 interconnecting thefree space propagation regions are additionally matched according to thegeneralised form of the design equations given above for non-arcuatecurves, it being understood that the waveguides of the array 142 willnot be simple arcuate curves in an AWG. The waveguides of the array 142are thus not only matched in terms of their optical path lengths, as ina conventional AWG, but additionally matched in terms of the beat lengthbetween the bend modes, according to the invention. In input waveguide140 couples into the first free space propagation region 148 and aplurality of output waveguides 145 couple out from the second free spacepropagation region 146. The waveguide array 142 has arranged thereon atrapezoidal electrode 144 biased in use by a voltage V for imposing alinear phase profile on the waveguide array 442, thereby to tune thedevice so that a given input wavelength from the input waveguide 140 canbe coupled to different ones of the output waveguides 145. The inclusionof the electrode 144 provides an active AWG. Omission of the electrodeis also possible to provide a passive AWG.

Ninth Embodiment

[0138]FIG. 22 is a schematic diagram of a mode converter according to anninth embodiment of the invention.

[0139] The above-described analysis of bent waveguides is not onlyapplicable to designing matched waveguide bends, but also permits thedesign of a mode converter.

[0140] As explained above, and also in reference [6], the field of amonomode bent waveguide can be described as the linear combination ofthe fundamental mode and the first leaky mode of the straight waveguide.By the same token, the field of a monomode straight waveguide can bedescribed as the linear combination of the fundamental mode and thefirst leaky mode of a bent waveguide. This can be mathematicallyexpressed by: $\begin{matrix}\left\{ \begin{matrix}{{\Theta_{0}\left( {x,y} \right)} = {{a_{00}{\Psi_{0}\left( {x,y} \right)}} + {a_{01}{\Psi_{1}\left( {x,y} \right)}}}} \\{{\Theta_{1}\left( {x,y} \right)} = {{a_{10}{\Psi_{0}\left( {x,y} \right)}} + {a_{11}{\Psi_{1}\left( {x,y} \right)}}}}\end{matrix} \right. & (8)\end{matrix}$

[0141] where Ψ_(0,1) are respectively the fundamental mode and the firstleaky mode of the straight waveguide, and Θ_(0,1) are respectively thefundamental mode and the first leaky mode of the bent waveguide. Thecomplex coefficients α_(ij) are the modal coefficients and are describedfurther in reference [6].

[0142] In FIG. 22, there is shown a single bend structure, generallysimilar to that of FIG. 4, comprising a first straight section 160followed by a curved section 158 (defined by parameters R, θ) and asecond straight section 155.

[0143] In the first straight waveguide section 160, an optical signal isshown propagating towards the bend, the signal having components of bothstraight modes Ψ₀ and Ψ₁. The bend is structured and dimensioned toprovide mode conversion so that, after the bend, i.e. at the output ofthe device, only the fundamental straight mode Ψ₀ is present. The deviceis thus a mode converter because it converts all the power of the Ψ₁mode to the Ψ₀ mode. Using the language of the previous embodiments, thebend is “unmatched” (but in a special way) in order to carry out therequired mode conversion. The output field Ψ_(OUT) can be written as

Ψ_(OUT)=APA⁻¹Ψ_(IN)SΨ_(IN)  (9)

[0144] where Ψ_(IN) is the vector of the input modes, A is the squarematrix with coefficients α_(ij) and P is the matrix describing thepropagation in the bent waveguide: $\begin{matrix}{P = {\left\lfloor \begin{matrix}{\exp \left( {{- j}\quad \beta_{B0}R\quad \theta} \right)} & 0 \\0 & {\exp \left( {{- j}\quad \beta_{B1}R\quad \theta} \right)}\end{matrix} \right\rfloor.}} & (10)\end{matrix}$

[0145] If the amplitudes and the phases of the input straight modes areknown, the radius R and the angle θ needed for the bend 158 of the modeconverter are found by imposing in equation (9) that $\begin{matrix}{\Psi_{OUT} = {\left\lfloor \begin{matrix}{\Psi_{0}\left( {x,y} \right)} \\0\end{matrix} \right\rfloor.}} & (11)\end{matrix}$

[0146] The mode converter is a useful device, since it is generallyimportant that substantially all the power of an optical signal is inthe fundamental mode. Power in the first leaky mode is undesirable,because of the radiative losses associated with that mode. The modeconverter can, for example, be used as a component of an opticalintegrated circuit. Suppose that at an arbitrary point along a waveguidein an optical integrated circuit, it can be predicted that a proportionof the optical signal power will be in the first leaky mode (e.g. at theoutput of a certain kind of device) and the phase is defined. It willthen be possible to insert a mode converter according to this embodimentto convert all the power back into the fundamental mode.

[0147]FIG. 23 shows an example of the use of a mode converter. Astraight waveguide section 160 is connected to an arbitrarily unmatchedbent waveguide section 158 with R=2 mm and θ=2.86°, connected in turn toa straight waveguide section 157 that is 150 μm long. (The waveguide isa ridge waveguide similar to that shown in FIG. 5 but without an uppercladding). The input field in the input straight section 160 is Ψ₀ andthe output field after the unmatched bend 158 in the straight section157 is 90.72% of Ψ₀ and 9.24% of Ψ₁.

[0148] However, if an appropriate mode converting bent waveguide isconnected to this structure, all the power can be reconverted to thefundamental mode only and the output field is undistorted. With thetechnique explained above (equations 9 to 11) the bent waveguide acts asmode converter. This has a bending radius of −300 μm (the sign minusmeans that the direction is opposite to the previous waveguide), θ=1.72°and its length is therefore only 9 μm. The total losses of the wholestructure are below 0.1%. The mode converter is very short.

[0149] In general, it is possible to design a mode converter using theabove-described design rules to convert an input mode with mixed Ψ₀ andΨ₁ contributions into an output mode purely with a Ψ₀ contribution,whatever the ratio between the strengths of the two modes Ψ₀ and Ψ₁.

[0150] Finally, it will be appreciated that a mode converter may be madeof multiple curved sections, not just the single curve shown.

Closing Remarks

[0151] All the powers (and intensities) mentioned in the abovedescription refer to the power of the first order (or fundamental) mode.Mathematically speaking, this power corresponds to the overlap integralbetween the transverse electric field present in the waveguide and thefield of the fundamental mode of the same waveguide. This is animportant point as all the techniques are based on the mode expansion ofthe waveguide field.

[0152] In all the examples given above, the waveguides are monomode. Theproposed approach including the design rule formulae stated above arealso valid if the waveguide is multimode in the vertical direction. Theprinciple of the invention is also still valid more generally formultimode waveguides. For bimodal waveguides, the closed-form formulaefor the optimum bend and offset are still valid. For other multimodewaveguides, the principles are still valid, but closed-form formulae donot exist.

References

[0153] [1] Neuman, Proc. IEE Journal, vol. 129, pages 278-280 (1982)

[0154] [2] Kitoh et. al., Journal of Lightwave Technology vol. 13, pages555-562 (1995)

[0155] [3] Hirono et al, IEEE Photonics Technology Letters vol. 10,pages 982-984 (1998)

[0156] [4] U.S. Pat. No. 5,243,672 (Dragone)

[0157] [5] Melloni et al, Proc. LEOS '99 pages 641-42 (1999)

[0158] [6] Melloni et al, Journal of Lightwave Technology, vol. 19,issue 4 (April 2001)

[0159] [7] Soldano, Journal of Lightwave Technology vol. 13, pages615-627 (1995)

[0160] These documents are referenced above and incorporated herein byreference in their entirety.

What is claimed is:
 1. A waveguide for guiding an optical fieldtherealong, the waveguide comprising a bend bounded by an input end andan output end, wherein the field in the bend is describable by first andsecond bend modes which are in phase at the input end but propagate atdifferent velocities through the bend, thereby coming out of phase andinto phase with each other in beats, wherein the bend is structuredhaving regard to its length and curvature to ensure that at the outputend the first and second bend modes are substantially in phase with eachother over a desired wavelength range of the optical field havingcompleted approximately an integer number of beats.
 2. A waveguideaccording to claim 1, wherein the first and second bend modes have aphase mismatch of less than one of 30, 20, 10, 5, 2 and 1 degrees at theoutput end of the bend over the desired wavelength range.
 3. A waveguideaccording to claim 1, wherein the bend has radius of curvature R that isvariable along the bend as a function of angle θ, where the bendsubstantially satisfies a matching condition: $\begin{matrix}{{\int_{{- \theta}/2}^{\theta/2}{\sqrt{{{R^{2}(\theta)}C_{1}} + C_{2}}\quad {\theta}}} = {2N\quad \pi}} \\{where} \\{{C_{1} = \left( {\beta_{0} - \beta_{1}} \right)^{2}};} \\{{C_{2} = {4\beta_{0}\beta_{1}c_{10}^{2}}};}\end{matrix}$

β₀ and β₁ are respective phase constants of first and second straightmodes; c₁₀ is a coupling coefficient indicative of coupling induced bythe bend between the first and second straight modes; and N is theinteger number of beats.
 4. A waveguide according to claim 1, whereinthe bend has at least one arcuate portion having radius of curvature Rand describing an angle θ, where the arcuate portion substantiallysatisfied a matching condition: $\begin{matrix}{R = \sqrt{\frac{\left( \frac{2N\quad \pi}{\theta} \right)^{2} - C_{2}}{C_{1}}}} \\{where} \\{{C_{1} = \left( {\beta_{0} - \beta_{1}} \right)^{2}};} \\{{C_{2} = {4\beta_{0}\beta_{1}c_{10}^{2}}};}\end{matrix}$

β₀ and β₁ are respective phase constants of first and second straightmodes: c₁₀ is a coupling coefficient indicative of coupling induced bythe bend between the first and second straight modes; and N is theinteger number of beats.
 5. A wavelength according to claim 3 or 4,wherein the matching condition is substantially satisfied when N iswithin one of 10%, 8%, 6%, 4%, 2%, 1%. 0.5% and 0.25% of an integervalue over the desired wavelength range.
 6. A waveguide according toclaim 1, wherein the bend comprises a plurality of curved portions.
 7. Awaveguide according to claim 6, wherein the bend comprises at least onesubstantially straight portion intermediate between adjacent curvedportions.
 8. A waveguide according to claim 6, wherein at least firstand second ones of the curved portions are oppositely curved.
 9. Awaveguide according to claim 8, wherein the bend provides a lateraloffset between two components aligned substantially in parallel witheach other.
 10. A waveguide according to claim 8, wherein the bendprovides a lateral offset between a substantially straight portion andfurther substantially straight portion that are substantially parallelto each other.
 11. A waveguide according to claim 8, wherein the bendprovides a lateral offset between a substantially straight portion and acomponent aligned substantially in parallel therewith.
 12. A waveguideaccording to any one of claims 9 to 11, wherein the substantiallystraight portion has no curve with a radius of curvature of less thanone of 20, 30, 40, 50 and 100 cm.
 13. A waveguide according to claim 6,wherein the plurality of curved portions comprise first and secondcurved portions having respective sets of parameters including shape anddimension, wherein the respective sets of parameters of the first andsecond curved portions differ.
 14. A Mach-Zehnder interferometer devicecomprising a first arm and a second arm, wherein at least the first armcomprises a waveguide according to claim
 1. 15. A waveguide branchcomprising an input waveguide and at least two output waveguides,wherein the input waveguide comprises a waveguide according to claim 1.16. A waveguide branch comprising an input waveguide and first andsecond output waveguides, wherein the first and second output waveguideseach comprise a waveguide according to claim
 1. 17. A multimodeinterference coupler having an input connected to a waveguide accordingto claim
 1. 18. A device comprising first and second waveguidesaccording to claim 1, that differ in their respective lengths and/orcurvatures.
 19. An arrayed waveguide grating in which at least some ofthe arms are waveguides according to claim
 1. 20. An arrayed waveguidegrating having an input connected to at least one waveguide according toclaim
 1. 21. A waveguide coupler comprising first and second waveguideswhich extend proximal to each other to form a coupling region havingfirst and second ends, wherein the first and second waveguides includebends according to claim 1, in order to diverge from each other at thefirst and second ends of the coupling region.
 22. An optical circuitcomprising an optical component having an input for receiving an opticalsignal and a waveguide according to claim 1, connected to the input. 23.An optical circuit according to claim 22, wherein the optical componentis of a type that is sensitive to mode distortion at the input.
 24. Amethod of manufacturing a waveguide for guiding an optical fieldtherealong, the waveguide comprising a bend bounded by an input end andan output end, the method comprising: describing the optical field inthe bend by first and second bend modes which are in phase at the inputend but propagate at different velocities through the bend, therebycoming out of phase and into phase with each other in beats; designingthe bend having regard to its length and curvature to ensure that at theoutput end the first and second bend modes are substantially in phasewith each other having completed approximately an integer number ofbeats over a desired wavelength range; and fabricating the waveguide.25. A mode converter comprising a waveguide for guiding a fieldtherealong, the waveguide having a bend bounded by an input end and anoutput end, wherein the bend is structured having regard to its lengthand curvature to convert an optical field received at the input end witha predictable ratio of a first power portion in a fundamental mode to asecond power portion in a first leaky mode into an optical field at theoutput end in which substantially all the power is in the fundamentalmode and substantially none in the first leaky mode.